Gravitational equilibrium in the presence of a positive cosmological constant

We reconsider the virial theorem in the presence of a positive cosmological constant {Λ}. Assuming steady state, we derive an inequality of the form ρ>~A(Λ/8πGN) for the mean density {ρ} of the astrophysical object. The parameter A depends only on the shape of the object. With a minimum at Aₛₚₕₑᵣₑ=2, its value can increase by several orders of magnitude as the shape of the object deviates from a spherically symmetric one. This indicates that flattened matter distributions such as, e.g., clusters or superclusters, with low density, cannot be in gravitational equilibrium.